As best I can work out, based on the record we have in Acts 9-15 and Gal 1 & 2, the sequence of events following Paul's conversion goes something like this. (The record in Acts is not continuous).
- Saul receives his conversion experience on the road to Damascus, Acts 9:1-19
- Immediately afterward Saul preached Jesus in the synagogues, Acts 9:20-22
- How long he remained in Damascus preaching in the synagogues in unknown but it was probably a few months(??). He then went to Arabia for an unknown time, presumably to clarify his thinking. Gal 1:17
- Saul then returned to Damascus. Acts 1:17. It is not clear whether the events of Acts 9:23-25 (the plot to kill Saul) was before or after the trip Arabia. I am inclined to the latter because it was after "many days" (Acts 9:23). In any case, at this point, Paul has thus far confined himself to synagogues. It was at this point that he decided to leave for Jerusalem.
- Paul goes to Jerusalem 3 years after his conversion (Gal 1:18, Acts 1:26-29) to meet Cephas/Peter and other leaders. Saul is still only dealing with Jews and not gentiles. Acts 9:29.
- Paul then travels to Syria and Cilicia via Tarsus. Gal 1:21-24, Acts 9:30.
- The Church enjoyed a time of peace. Acts 9:31. During this next period, Acts is silent about the activities of Saul.
- Peter miraculously heals Aeneas and resurrects Tabitha at Lydda. Acts 9:32-43
Peter is called to Cornelius and baptises the first gentile at Caesarea. Acts 10.
Peter is called to Jerusalem to explain why he had entered a gentile's house and then baptised a gentile. After explaining about the action of the Holy Spirit, the leaders had no further objection. Acts 11:1-18. There is no further record of Peter ever preaching to gentiles after this incident.
- Attention then shifts to Antioch and the developing tension over the role, if any, gentiles should have in the Gospel message. Acts 11:19-26. It was during this time that Barnabas went to get Saul in Tarsus (v25) and brought him to Antioch. It was at this time that Saul and Barnabas began preaching to gentiles.
- Because of a famine, Saul and Barnabas are sent with gifts to Jerusalem. Acts 11:27-30. It is probable that this was trip to Jerusalem is that referred to in Gal 2:1, fourteen years after Paul's conversion. It was at this Jerusalem meeting that Paul agreed with Peter an amicable division of labour - Paul would preach to the gentiles while Peter would preach to the Jews, Gal 2:9.
- Saul returns from Jerusalem to Antioch. Acts 12:25
- Saul/Paul and Barnabas are sent off on their first missionary journey Acts 13:2 - 14:28, preaching to gentiles, returning to Antioch.
- The Jerusalem council is called to debate circumcision. Acts 15.
Thus, the "three years" (Gal 1:18) appears to be three years after Paul's conversion, and NOT the time he spent in either Damascus or Arabia.
Ellicott reaches the same conclusion:
(18) After three years.—This date is probably to be reckoned from the
great turning-point in the Apostle’s career—his conversion. It need
not necessarily mean three full years, just as the three days during
which our Lord lay in the grave were not three full days. It may have
been only one whole year and parts of two others; but the phrase may
equally well cover three whole years. This ambiguity shows the
difficulty of constructing any precise system of chronology.